To determine which equation represents a line that is perpendicular to the line passing through the points [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex], we follow these steps:
### Step 1: Calculate the Slope of the Line Passing Through the Points
The slope (m) of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is determined by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given points:
[tex]\[
(x_1, y_1) = (-4, 7)
\][/tex]
[tex]\[
(x_2, y_2) = (1, 3)
\][/tex]
Substitute the points into the slope formula:
[tex]\[ m = \frac{3 - 7}{1 + 4} = \frac{-4}{5} \][/tex]
So, the slope of the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is:
[tex]\[ m = -\frac{4}{5} \][/tex]
### Step 2: Determine the Slope of the Perpendicular Line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. The negative reciprocal of [tex]\(-\frac{4}{5}\)[/tex] is:
[tex]\[ m_{\text{perp}} = -\frac{1}{-\frac{4}{5}} = \frac{5}{4} \][/tex]
### Step 3: Find the Equation of the Perpendicular Line
We need the equation of the line with slope [tex]\(\frac{5}{4}\)[/tex]. We are given four possible choices:
A. [tex]\( y = \frac{5}{4} x + 8 \)[/tex]
B. [tex]\( y = \frac{4}{5} x - 3 \)[/tex]
C. [tex]\( y = -\frac{4}{5} x + 6 \)[/tex]
D. [tex]\( y = -\frac{5}{4} x - 2 \)[/tex]
The slope identified in Step 2 for the perpendicular line is [tex]\(\frac{5}{4}\)[/tex], which matches the slope of the line in option A.
### Conclusion
The equation of the line that represents a line perpendicular to the line passing through [tex]\((-4, 7)\)[/tex] and [tex]\((1, 3)\)[/tex] is:
[tex]\[ \boxed{y = \frac{5}{4} x + 8} \][/tex]
